Discrete invariants of varieties in positive characteristic. (English) Zbl 1084.14023

Let \(F: X \to S\) be a smooth proper morphism of sheaves in characteristic \(p >0\). If the sheaves \(R^b f_* \Omega^a_{X/S}\) are locally free and the Hodge-de Rham spectral sequence degenerates at \(E_1\), then the de Rham cohomology sheaves \(\mathcal M = H^m_{\text{dR}} (X/S)\) are locally free \(\mathcal O_S\)-modules equipped with a descending Hodge filtration \(C^\bullet\) and an ascending conjugate filtration \(D_\bullet\) as well as with \(\mathcal O_S\)-linear isomorphisms \(\varphi_i: (\text{gr}^i_C)^{(p)} \to \text{gr}_i^D\) given by the inverse Cartier operator. In this paper the authors give a complete classification of the structures \((\mathcal M, C^\bullet, D_\bullet, \varphi_\bullet)\) over an algebraically closed field. The result shows that such structures are essentially combinatorial objects.


14F40 de Rham cohomology and algebraic geometry
14C30 Transcendental methods, Hodge theory (algebro-geometric aspects)
14G35 Modular and Shimura varieties
14J28 \(K3\) surfaces and Enriques surfaces
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